Let \(X\) and \(Y\) be Banach spaces and \(m\) be a positive integer. A (continuous) mapping \(P: X\to Y\) is said to be an \(m\)-homogeneous polynomial if there is a (continuous) symmetric \(m\)-linear mapping \({\buildrel\vee\over P}: X\times X\times\dots\times X\to Y\) such that \(P(x)={\buildrel\vee\over P}(x,\dots,x)\) for all \(x\) in \(X\). An \(m\)-homogeneous polynomial \(P\) on \(E\) is said to be strongly \(p\)-summing if there is \(C>0\) such that, for all \(x_1,\dots, x_n\) in \(X\), \[ \left(\sum_{j=1}^n\|P(x_j)\|^p\right)^{1/p}\leq C \sup_{\Phi\in B_{{\mathcal P} (^mX)}}\left(\sum_{j=1}^\infty|\Phi(x_j)|^p\right)^{1/p}, \] \(p\)-dominated if there is \(C>0\) such that, for all \(x_1,\dots, x_n\) in \(X\), \[ \left(\sum_{j=1}^n\|P(x_j)\|^{p/m}\right)^{m/p}\leq C \sup_{\phi\in B_{X'}}\left(\sum_{j=1}^n|\phi(x_j)|^p\right)^{m/p}, \] and Cohen strongly \(p\)-summing if there is \(C>0\) such that, for all \(x_1,\dots, x_n\) in \(X\), all \(y_1^*,\dots, y_m^*\) in \(Y'\), \[ \sum_{j=1}^n|\langle P(x_j),y_j^*\rangle|\leq C \left(\sum_{j=1}^n\|x_i\|^{mp} \right)^{1/p}\sup_{y\in B_{Y}}\|y_j^*(y)\|_{l_p^n}. \] An analogous definition is made for a Cohen strongly \(p\)-summing \(m\)-linear mapping. The authors show that an \(m\)-homogeneous polynomial \(P: X\to Y\) is Cohen strongly \(p\)-summing if and only if its associated \(m\)-linear mapping, \(\buildrel\vee\over P\), is Cohen strongly \(p\)-summing if and only if its linearization \(\tilde P: {\widehat{\mathstrut \bigotimes\atop {s,m,\pi}}}X \to Y\) is \(p\)-dominated. It is shown that, if \(m\in {\mathbb N}\) and \(1